MultiplicityOfSmallestPart

How many parts equal the smallest.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000160via IntegerPartition✓↗St001457via PlanePartition✓↗

MultiplicityOfSmallestPart(partition)How many parts equal the smallest.

Typed signatures

Type	Package	Applies
(integer_partition | plane_partition) -> number	combinatorics	always

Domain: Partition statistics

Details

• Defined over IntegerPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerPartition and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

0 if Length(_x) < 1 else Count(Filter(
                                       Range(1, Length(_x)),
                                       Function(_x[i] == _x[Length(_x)], i),
                                     ))

on PlanePartition, over _x

0 if Length(Flatten(_x)) < 1 else Apply((reading: list<integer>) => Apply(
                                                                           (lengths: list<integer>) => Apply(
                                                                                                              (least: integer) => Max(Sum(Map(Function(1 if e == least else 0, e), reading)), 0),
                                                                                                              Min(reading),
                                                                                                            ),
                                                                           Map(
                                                                                Function(Length(r), r),
                                                                                _x,
                                                                              ),
                                                                         ), Flatten(_x))

Examples

Basic #

the partition $4 + 2 + 1$ of $7$

live ↗

In$\mathrm{MultiplicityOfSmallestPart}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$1$

the partition $3 + 3 + 1$ of $7$

live ↗

In$\mathrm{MultiplicityOfSmallestPart}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$1$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([1]), \text{MultiplicityOfSmallestPart})$Out$1$

Known value · FindStat St000160

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{MultiplicityOfSmallestPart})$Out$1$

Known value · FindStat St000160

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{MultiplicityOfSmallestPart})$Out$1$

Known value · FindStat St000160

About

Examples	5
Crosswalk	FindStat ×2